ARTICLE IN PRESS
Deep-Sea Research I 55 (2008) 277–295
www.elsevier.com/locate/dsri
Thin layers of plankton: Formation by shear
and death by diffusion
Daniel A. Birch, William R. Young, Peter J.S. Franks
Scripps Institution of Oceanography, University of California at San Diego, La Jolla, CA 92093-0213, USA
Received 22 June 2007; received in revised form 15 November 2007; accepted 19 November 2007
Available online 23 December 2007
Abstract
We show that a steady vertically-sheared current can produce a thin layer of plankton by differentially advecting an
initial patch whose vertical and horizontal dimensions are H 0 and L0 , respectively. Our model treats the plankton as an
inert passive tracer with vertical diffusivity kv and subject to a vertically-sheared horizontal current with shear a. After a
transient of duration L0 =aH 0 the vertical thickness H of the patch decreases with HðtÞ L0 =at. This shear-driven thinning
1=3 2=3
is halted by diffusion at a time of order a2=3 kv L0 , and at this time the layer achieves a minimum layer thickness of
1=3 1=3 1=3
order a
kv L0 . For typical oceanic parameters, such as kv 105 m2 s1 , a102 s1 , and L0 1000 m the initial
transient is about 3 h and the layer achieves a minimum thickness of order 1 m in a time of order 1 day. During the shear
thinning the intensity of the layer decreases by a factor of 31=2 0:58, which means that the intensity of the thin layer is
comparable to the intensity of the patch from which it was formed. Subsequently the layer thickens and its intensity
decreases; the coup de grace is delivered by shear dispersion at a time of order H 20 =kv . The lifetime of the thin layer, defined
by the condition that the maximum concentration is comparable to the initial maximum concentration, is the same order as
the time to achieve minimum thickness. Additionally, analysis of a nutrient–phytoplankton model shows that
phytoplankton growing in a sheared patch of nutrients can result in a layer of phytoplankton that develops as an
initially thin feature.
r 2007 Elsevier Ltd. All rights reserved.
PACS: 92.20.Jt; 92.10.Ns; 87.23.Cc; 92.40.Cy
Keywords: Thin layers; Plankton; Finestructure; Advection–diffusion; Patchiness; Mathematical models
1. Introduction
Thin layers of phytoplankton appear to be
ubiquitous features of coastal waters (Strickland,
Corresponding author. Tel.: +1 858 534 1504;
fax: +1 858 534 8045.
E-mail addresses: dbirch@ucsd.edu (D.A. Birch),
wryoung@ucsd.edu (W.R. Young),
pfranks@ucsd.edu (P.J.S. Franks).
0967-0637/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.dsr.2007.11.009
1968; Derenbach et al., 1979; Jaffe et al., 1998;
Dekshenieks et al., 2001; Johnston et al., 2008).
Discovered by Strickland (1968), thin layers were
first sampled in detail by Derenbach et al. (1979)
using one of the earliest in situ fluorometers.
Phytoplankton thin layers generally appear as
spikes in vertical profiles of fluorescence or beam
attenuation; these spikes are often many times the
background intensity and have vertical scales of tens
of centimeters to meters. More recently, the
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278
Vertical distance (cm)
individual plankters forming a thin layer have been
imaged by Franks and Jaffe (2008) (Fig. 1).
Because thin layers have usually been identified
using vertically profiling instruments, their horizontal extent is not well known. Derenbach et al. (1979)
suggested a horizontal scale of 10 m for the layers
they observed using a profiling fluorometer coupled
with diver observations. Franks (1995) hypothesized
that along-isopycnal patch scales of o1 km would
generate thin layers through shearing by nearinertial waves. Dekshenieks et al. (2001) and
McManus et al. (2003) showed that thin layers of
phytoplankton fluorescence had horizontal scales of
5 km or less in a long, narrow fjord.
Recently Franks and Jaffe (2008) used an imaging
fluorometer system and found thin layers of cell types
(defined by their shape and size) that did not appear
as layers of enhanced biomass or fluorescence. Thin
layers of zooplankton have also been observed using
visual (Jaffe et al., 1998) and acoustic techniques
(Holliday et al., 2003; McManus et al., 2005).
Alldredge et al. (2002) observed a thin layer of
0
0
5
5
10
10
15
15
20
20
25
25
30
30
35
35
40
marine snow in a vertical profile of closely spaced
photographs.
Thin layers are a significant source of small-scale
vertical heterogeneity in the marine environment.
Herbivorous zooplankton are known to have
behaviors that enable them to exploit patches of
phytoplankton by turning more frequently in
regions of higher phytoplankton concentration
and swimming straighter in regions of lower
phytoplankton concentration—this results in the
zooplankton spending more time where there is
more phytoplankton to graze on (e.g., Tiselius,
1992). Thus thin layers of high biomass may be sites
of significantly enhanced trophic coupling. Highbiomass layers will have smaller nearest-neighbor
distances, with potentially greater competition,
nutrient recycling, infection, grazing, and sexual
exchange than the surrounding waters. Dense
concentrations of certain species, for example, a
toxic alga (Rines et al., 2002) may actually provide a
refuge from grazing pressures. Understanding the
dynamics that lead to thin-layer formation will
40
0
5
10
Distance (cm)
0
5
10
15
Cells/ml
Fig. 1. An example of a thin layer imaged in situ with a planar laser imaging fluorometer (Franks and Jaffe, 2008). The left panels show a
sequence of images spanning a 40 cm vertical distance. The images show the chlorophyll a fluorescence of individual plankters. There is a
gap between the images from 21.5 to 26 cm. The right panel shows fluorescent particle counts in 1 cm vertical bins. The solid line is a
Gaussian fit to the particle counts.
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D.A. Birch et al. / Deep-Sea Research I 55 (2008) 277–295
increase our understanding of the dynamics structuring marine ecosystems and the biogeochemical
fluxes they mediate.
Despite their apparent ubiquity, the dynamics
of thin layers are not well understood. Both physical
and biological mechanisms for the formation of
thin layers have been proposed; the actual dynamics
are probably a combination of physical and
biological factors whose importance varies with
time and space. Examples of biological processes
that may influence the formation of thin layers
include swimming, growth, the inhibition of grazing, and the increase in local viscosity by exudates.
Swimming is definitely important to some kinds
of thin layers, such as those formed by dinoflagellates and zooplankton (e.g., Rasmussen and
Richardson, 1989; McManus et al., 2005). Reproduction may be important to some kinds of thin
layers, especially those formed during blooms
(Nielsen et al., 1990). Zooplankton may avoid
certain kinds of phytoplankton layers, and this
may help maintain those layers (Fiedler, 1982).
Finally, phytoplankton may also increase the local
viscosity and therefore decrease the turbulent
diffusion of thin layers (Jenkinson and Biddanda,
1995).
While biological dynamics are undoubtedly important in thin-layer formation, it is essential to
fully understand the role of physical mechanisms.
Physical processes are generally more easily measured in the field, and provide a useful null
hypothesis for layer formation. Layer formation
by sinking particles decelerating through a pycnocline has been proposed as a mechanism for the
formation of phytoplankton (Derenbach et al.,
1979) and marine snow (Alldredge et al., 2002)
layers. Franks (1995) explored the creation of thin
layers of plankton by the vertical shear of nearinertial internal waves. Because of the oscillating
nature of these waves, the layers shown by Franks
were transient reversible features which disappeared
after a full inertial period. Johnston et al. (2008)
examined thin layers formed by shear in the
transition zone during an upwelling relaxation
event.
Here we investigate the simplest physical mechanism which might be responsible for the formation of
thin layers: Eckart (1948) noted that verticallysheared horizontal currents create sharp vertical
property gradients from weak horizontal gradients
(see also Osborn, 1998). This fact is the basis of
our hypothesis that vertically-sheared horizontal
279
currents are important in forming thin layers.
Building on the work of Eckart (1948) and Novikov
(1958) we explore the effects of steady shear and
diffusion on the formation of a thin layer from an
initial patch of plankton. Several of the results
contained in this paper are also found in Young
et al. (1982), Rhines and Young (1983) and Stacey
et al. (2007). We obtain these results in the context
of an explicit model for the formation of thin layers
whereas Rhines and Young (1983) consider mixing
of a passive tracer and Stacey et al. (2007) begins
with an equation for the evolution of an abstract
layer thickness.
Section 2 describes a simple physical model for
the formation of thin layers and presents analysis of
the layer properties. The mathematical details of
this analysis are contained in Appendix A. Section 3
adds nutrients and plankton growth to our physical
model and Appendix B contains the details of the
analysis of the nutrient–phytoplankton (NP) model.
Section 4 contains a summary and discussion of our
results.
2. The simplest physical model of thin layers
The basis of this work is a kinematic model
showing that a steady vertical shear thins an
initial patch of plankton with large vertical scale
into a layer that is thin in the vertical (Fig. 2).
The model includes diffusion, which tends to
counteract the shear-driven layer thinning by
dispersing the plankton. Diffusion also decreases
the maximum plankton concentration in the
patch. Initially we will treat the plankton as an
inert passive tracer and ignore all biological
processes. The differential equation describing this
model is
Pt þ azPx ¼ kh Pxx þ kv Pzz .
(1)
Pðx; z; tÞ is the concentration of plankton at the
horizontal position x, vertical position z, and time t.
Subscripts of P denote differentiation with respect
to the subscripted variable. The parameters are the
vertical shear a, the horizontal diffusivity of
plankton kh , and the vertical diffusivity of plankton
kv . The variables and their units are summarized in
Table 1.
We take as an initial condition a pre-existing
patch of plankton with finite horizontal and vertical
scales and a concentration that is highest at the
center and tapers off toward the edges of the patch.
This condition can be modeled as a Gaussian
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20
20
t=0
z (m)
z (m)
10
0
–10
–20
–2000 –1000
20
0
x (m)
1000
0.9
0
0.8
–10
0.7
20
t = (tshear tmin)1/2
z (m)
10
z (m)
10
–20
–2000 –1000
2000
0
2000
0.5
0.4
0.3
0
0.2
0.1
–20
–2000 –1000
–20
–2000 –1000
2000
1000
10
–10
1000
0.6
0
x (m)
t = tmin
–10
0
x (m)
1
t = tshear
P (arbitrary units)
280
0
0
x (m)
1000
2000
Fig. 2. Solutions of the model in Eq. (1) with the initial condition in Eq. (2). (a) The initial distribution of plankton. (b) The distribution at
the end of the tilting phase ðtshear ¼ 104 sÞ. (c) The distribution of plankton midway through the shear-thinning phase (specifically, at the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
geometric mean of tshear and tmin : t ¼ tshear tmin 11 h). (d) The distribution at the time of minimum thickness ðtmin 42 hÞ. The
parameters are L0 ¼ 1000 m, H 0 ¼ 10 m, kh ¼ 1 m2 s1 , kv ¼ 105 m2 s1 , and a ¼ 102 s1 . The color scale for P is the same in all panels
and the units are arbitrary.
Table 1
The variables and their units for the constant-shear model in Eq.
(1) with the Gaussian initial condition in Eq. (2)
Pðx; z; tÞ
x
z
t
a
kh
kv
P
L0
H0
HðtÞ
LðtÞ
SðtÞ
IðtÞ
tmin
H min
tshear
tdiff
Concentration of plankton (e.g., mg Chl m3 )
Horizontal coordinate (m)
Vertical coordinate (m)
Time (s)
Vertical shear of the horizontal velocity ðs1 Þ
Horizontal diffusivity of plankton ðm2 s1 Þ
Vertical diffusivity of plankton ðm2 s1 Þ
Total plankton in the xz-plane (e.g., mg Chl m1 )
Initial horizontal extent of the plankton layer (m)
Initial vertical thickness of the plankton layer (m)
Vertical thickness of the plankton layer (m)
Horizontal extent of the plankton layer (m)
Slope of the layer (dimensionless)
Layer intensity (dimensionless)
Time of minimum layer thickness (s)
Minimum layer thickness (m), H min ¼ Hðtmin Þ
Shear time (s)
Diffusion time (s)
distribution of plankton concentration:
!
P
x2
z2
exp 2 Pðx; z; 0Þ ¼
.
2pL0 H 0
2L0 2H 20
(2)
The model in Eqs. (1) and (2) assumes that the
plankton concentration is uniform in the second
horizontal dimension. The total amount of plankton in the patch in the xz-plane in Eq. (2) is P. The
maximum plankton concentration, located at
ðx; zÞ ¼ ð0; 0Þ, is P=ð2pL0 H 0 Þ, and the initial vertical
and horizontal length scales are H 0 and L0 . The
patch could represent a region of enhanced biomass
formed by some previous event, or it could be a
patch of a certain plankton type or community that
is distinct from the surrounding community.
We seek solutions of Eqs. (1) and (2) in order to
understand the dependence of the layer thickness,
extent, and intensity on time, the initial conditions,
and the physical parameters (shear and diffusivity).
Our main results concerning the smallest possible
vertical scale [e.g., the scaling of the minimum layer
thickness in Eq. (21)] do not depend on the
particular initial condition in Eq. (2), provided that
the initial condition has a finite horizontal scale L0 .
However, the Gaussian initial condition, Eq. (2), is
convenient because it has an analytic solution
(derived in Appendix A and illustrated in Fig. 2).
This explicit solution enables us to thoroughly
explore all aspects of the advection–diffusion model
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in Eq. (1) and quantitatively understand sheardriven layer formation and dissipation.
The exact solution for Pðx; z; tÞ given in Appendix
A relies on the observation that the plankton
distribution maintains its Gaussian structure for
all time. Preservation of Gaussianity is a venerable
result due to Novikov (1958), who considered a
d-function initial condition. Other work has mainly
focused on horizontal shear dispersion (Taylor,
1953; Okubo, 1968; Wilson and Okubo, 1978;
Young et al., 1982; Rhines and Young, 1983;
Sundermeyer and Ledwell, 2001). Here, instead,
we explore the dynamics of the intensity and vertical
thickness of layers; these are the properties that are
commonly observed with vertically profiling instruments. Our use of the initial condition in Eq. (2),
which is a Gaussian patch rather than a d-function,1
introduces two new length scales: H 0 and L0 . Our
goal is to explain how thin layers form by showing
how a patch with initial thickness H 0 is transformed
into a layer with thickness HðtÞ5H 0 .
2.1. Definition of layer thickness HðtÞ, extent LðtÞ,
slope SðtÞ, and intensity IðtÞ
Every vertical profile through our model patch
Pðx; z; tÞ turns out to be a Gaussian whose standard
width is independent of x. Therefore, we define the
layer thickness HðtÞ to be the standard width of the
Gaussian profile observed in any vertical profile.
Using this definition of the layer thickness we find
that the layer thickness of the initial condition
Pðx; z; 0Þ in Eq. (2) is H 0 . The formal definition of
HðtÞ for profiles of any form is given in Eq. (A.16).
To quantify the horizontal extent of a layer we
use the root mean square length LðtÞ defined in
Eq. (A.31), which equals the maximum distance in
the horizontal from the center of the patch at which
the plankton can be detected at a significant fraction
[expð 12Þ 61% in our Gaussian model] of the peak
plankton concentration at any depth in a vertical
profile. LðtÞ is the traditional measure of horizontal
scale employed in the shear-dispersion literature
(e.g., Sundermeyer and Ledwell, 2001). The analogous quantity in the vertical, the total depth range
occupied by the plankton, is much greater than HðtÞ
(Fig. 3).
A signature characteristic of thin layers created by
vertical shear is that the layers are tilted across
1
The special case of a d-function initial condition can be
recovered as the limit L0 ! 0 and H 0 ! 0.
281
isopycnals. We define SðtÞ to be the slope one finds
by drawing a straight line through the maxima of P
in each vertical profile. SðtÞ is different from the slope
of the major axis of an elliptical contour of constant
P; if one examines a set of vertical profiles through a
phytoplankton patch, it is easier to estimate SðtÞ than
to locate the major axis of the ellipse.
In addition to the size and shape of the layer, we
are also interested in the layer intensity IðtÞ, which
we define as the ratio of the maximum concentration of plankton divided by the maximum concentration of plankton in the initial condition, Eq. (2).
Because of the symmetry of the initial condition, the
maximum concentration at any time is Pð0; 0; tÞ and
therefore the definition of I is equivalent to
IðtÞ Pð0; 0; tÞ
.
Pð0; 0; 0Þ
(3)
2.2. Weak diffusion
Our discussion will focus on the weak-diffusion
regime which is characteristic of oceanic conditions
and most relevant to layer formation. Diffusion is
weak in the sense that the time to diffuse through L0
and H 0 is much longer than the advective time scale.
To estimate the advective time scale we argue that
the differential horizontal velocity on two streamlines separated by H 0 is aH 0 . Therefore the time to
shear the initial patch by differential horizontal
advection through L0 is
tshear L0
,
aH 0
(4)
called tstart by Stacey et al. (2007). Forming the ratio
of the vertical diffusion time,
tdiff H 20
,
kv
(5)
to tshear , we see that in order for diffusion to be weak
we must have
aH 30
b1.
L0 kv
(6)
Comparing the horizontal diffusion time, L20 =kh , to
tshear leads to a second requirement for weak
diffusion, namely
aH 0 L0
b1.
kh
(7)
The non-dimensional combinations above are the
Péclet numbers for our model. The assumption of
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282
20
15
P(-2.5L0, z, t)
10
P(0, z, t)
z (m)
5
2H(t)
0
≈2H0
≈2L0
-5
2H(t)
-10
2L(t)
-15
-20
-5000
-4000
-3000
-2000
-1000
0
x (m)
1000
2000
3000
4000
5000
Fig. 3. A schematic of a plankton layer illustrating the definitions of layer thickness HðtÞ and extent LðtÞ. Note that the layer thickness
HðtÞ is determined by the standard deviation of the Gaussian profiles, not by the distance between any particular contours of constant P.
The initial dimensions, L0 and H 0 , are also shown. The total depth range is indicated as 2H 0 because in this illustration t5tdiff , where
tdiff is the vertical diffusion time scale defined in Eq. (5)—there has not been time for diffusion to transport plankton outside of the initial
depth range.
large Péclet number may be justified by considering
the results of the coastal mixing and optics (CMO)
experiment. One of the major objectives of the
CMO experiment was to understand mixing in the
vertical over the continental shelf (Dickey and
Williams, 20012), which is what dissipates the thin
layers in our model. As part of the CMO,
MacKinnon and Gregg (2003) found diapycnal
diffusivities between 5 106 and 2 105 m2 s1
over the New England shelf late in the summer of
1996. Ledwell et al. (2004) and Oakey and Greenan
(2004) found diapycnal diffusivities between 106
and 105 m2 s1 in the same place, but not at
exactly the same time. During the spring of 1997
MacKinnon and Gregg (2005) found much greater
diapycnal diffusivities (4103 m2 s1 ), but also subcritical Richardson numbers, which have been
observed to be inconsistent with thin layers
(Dekshenieks et al., 2001; McManus et al., 2005).
Also as part of the CMO experiment, Sundermeyer
and Ledwell (2001) found horizontal diffusivities
2
Volume 106 (2001) of the Journal of Geophysical Research:
Oceans is a special issue dedicated to the CMO experiment.
between 0.3 and 4:9 m2 s1 . Observed shears during
the CMO experiment were of order 102 s1
(MacKinnon and Gregg, 2003; Ledwell et al.,
2004; Oakey and Greenan, 2004). Given the results
of the CMO experiment, we choose kh ¼ 1 m2 s1 ,
kv ¼ 105 m2 s1 , and a ¼ 102 s1 as typical values.
Using our chosen values for kh , kv , and a the
conditions in (6) and (7) are satisfied for oceanic
parameter values if H 0 \10 m and L0 \100 m.
2.3. The four evolutionary phases
In the case of large Péclet number, the evolution
of the plankton distribution obtained from Eqs. (1)
and (2) consists of four phases (Fig. 2 shows the first
two):
(1) The tilting phase during which the initial
distribution is tilted and stretched with little
change in HðtÞ or IðtÞ.
(2) The shear-thinning phase during which HðtÞ
decreases monotonically, with HðtÞ / t1 , until
a minimum layer thickness H min is reached at
t ¼ tmin .
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283
(3) The decay phase during which HðtÞ increases
slowly due to vertical diffusion and IðtÞ
decreases to very low levels.
(4) The shear-dispersion phase, beginning at t ¼ tdiff ,
when HðtÞ has finally grown back to the initial
thickness H 0 .
approximations for the layer thickness and slope
through the shear-thinning phase.
The expression in Eq. (10) shows that for every x
the vertical profile of Pðx; z; tÞ has a Gaussian form
with thickness HðtÞ, centered about the depth
z ¼ SðtÞx. In other words, the developing layer is
tilted along the line
Diffusion is negligible during the tilting phase and
throughout most of the shear-thinning phase. The
end of the shear-thinning phase is reached when the
layer is so thin that the weak vertical diffusion is
finally able to compete with differential advection.
During the decay phase IðtÞ decreases as t3=2 as the
layer erodes via horizontal processes. During the
final shear-dispersion phase the intensity is very
small, and decreases as t2 . We now discuss these
four phases in some quantitative detail.
z ¼ SðtÞx.
2.4. The tilting and shear-thinning phases
The assumption that diffusion is very weak allows
the evolution of Eq. (1) during the tilting and shearthinning phases to be approximated by neglecting
diffusion
Pt þ azPx ¼ 0.
(8)
The simplified model in Eq. (8) may be solved by the
method of characteristics, giving the solution
Pðx; z; tÞ ¼ Pðx azt; z; 0Þ.
(9)
With the Gaussian initial condition in Eq. (2), the
advective solution in Eq. (9) can be put into the
form
P
x2
½z SðtÞx2
Pðx; z; tÞ ¼
exp 2 ,
2pL0 H 0
2L ðtÞ
2H 2 ðtÞ
(10)
where
L0 H 0
HðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
2
L0 þ H 20 a2 t2
LðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L20 þ H 20 a2 t2 ,
(11)
The slope SðtÞ in Eq. (13) increases from zero and
achieves a maximum S max ¼ H 0 =ð2L0 Þ at time tshear
defined in Eq. (4). We define the tilting phase to be
when
0ptptshear .
tshear ptptmin .
L20
H 20 at
.
þ H 20 a2 t2
(13)
The non-diffusive equation for LðtÞ in Eq. (12) is a
very good approximation to the full solution
through the decay phase (the difference is indistinguishable in Fig. 4). Eqs. (11) and (13) are good
(16)
The time tmin is estimated below in Eq. (22), and if
diffusion is weak then tmin btshear , so that the shearthinning phase lasts much longer than the tilting
phase.
During the beginning of the shear-thinning phase
the layer thickness in (11) can be approximated by
L0
,
at
(Fig. 5) and the slope in Eq. (13) becomes
(17)
1
.
(18)
at
We emphasize that the slope of the layer is
independent of the initial parameters L0 and H 0 ,
and that the layer thickness HðtÞ is independent of
the initial layer thickness H 0 . Instead, the initial
SðtÞ SðtÞ (15)
During the tilting phase the patch is distorted by
differential advection with little change in HðtÞ. At
the end of the tilting phase, when t ¼ tshear
pffiffiffi, the layer
thickness HðtÞ is approximately3 H 0 = 2 0:7H 0 .
The maximum concentration of plankton in the
layer at the end of the tilting phase is still
indistinguishable from the initial maximum concentration (see Appendix A.2 for details). The duration
of the tilting phase is determined by the initial patch
size and the strength of the shear; for shears less
than 102 s1 , an initial patch L0 ¼ 1 km wide by
H 0 ¼ 10 m thick tilts, with little thinning, for 3 h or
more.
The shear-thinning phase begins when t ¼ tshear ,
and continues until the minimum layer thickness
H min is achieved at t ¼ tmin . Thus the shear-thinning
phase occurs when
HðtÞ (12)
(14)
3
This estimate follows from inserting (4) into (11).
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284
101
H (m)
Tilting
Shear-thinning
tshear
Decay
tmin
100
100
101
t (hours)
102
L (m)
105
Tilting
Shear-thinning
Decay
104
tmin
tshear
103
100
Intensity
100
101
t (hours)
Tilting
102
Shear-thinning
Decay
10-1
tmin
tshear
10-2
100
101
t (hours)
102
Fig. 4. The layer thickness, extent, and intensity as a function of time. The final time on all three plots is 10tmin and the parameters are
L0 ¼ 1000 m, H 0 ¼ 10 m, kh ¼ 1 m2 s1 , kv ¼ 105 m2 s1 , and a ¼ 102 s1 . The vertical dashed lines are located at tshear , the end of the
tilting phase defined in Eq. (4), and tmin , the end of the shear-thinning phase. The shear-dispersion phase occurs for tb10tmin and may be
seen in Fig. 6. (a) The layer thickness as a function of time. The ’s are the approximations for layer thickness at the end of the tilting phase
in Eq. (11) and at the approximate time of minimum layer thickness in Eq. (A.28). The dashed line is the approximation to HðtÞ in
Eq. (26). (b) The layer extent as a function of time; the dashed line is the approximation to LðtÞ in Eq. (12) and is indistinguishable from the
solid line. (c) The layer intensity as a function of time. The dashed line is the approximation to the intensity in Eq. (23) using the
approximation for tmin in Eq. (A.26).
layer width L0 determines the thickness of the layer
in Eq. (17).
Both the layer thickness and slope decrease with
time as ðatÞ1 during the shear-thinning phase. The
approximation in Eq. (17) works because the layer
thickness is determined by the vertical separation
between the upstream and downstream edges of the
patch of plankton-rich water. The difference between
the distances travelled by two plankters, one on each
edge of the patch and having the same x coordinate,
is 2Hat, where 2Ha is the difference of the velocities
of the two plankters and t is the elapsed time. This
difference in distance travelled must equal the
distance a particle from the upstream edge has to
travel to catch up with a particle from the downstream edge, which is approximately 2L0 (Fig. 5).
The relationship in Eq. (17) is only approximately
true because with the initial condition in Eq. (2), the
upstream and downstream edges of the patch are not
perfectly parallel and Eq. (17) does not include
diffusion. Stacey et al. (2007) contains an alternative
explanation of this t1 thinning based on the angle of
the layer with the horizontal.
2.5. Minimum layer thickness
If kv 40, then the shear thinning of the layer
cannot continue indefinitely; eventually the layer
becomes so thin, and the vertical gradient of P so
large, that diffusion halts the thinning regime in
Eq. (17). This signifies the end of the shear-thinning
phase and occurs at t ¼ tmin . Here we use simple
scaling arguments to estimate tmin , H min , and the
intensity Iðtmin Þ.
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z (m)
D.A. Birch et al. / Deep-Sea Research I 55 (2008) 277–295
285
(z1+2H)αt
z1+2H
2L0
2H(t)
z1αt
z1
0
–L0
L0 + z1αt
L0
0
x (m)
Fig. 5. A schematic of the shear thinning of the advection-only model in Eq. (8) with the initial condition in Eq. (2). As time increases the
initial patch rotates and stretches. The distance in the vertical between the two points indicated by the ’s is 2HðtÞ. The initial locations of
the points are indicated by the ’s. The difference of the distances travelled by the two points is 2Hat, which is also 2L0 —this can be seen in
the figure where L0 þ z1 at ¼ L0 þ ðz1 þ 2HÞat.
The minimum layer thickness occurs when shear
thinning (represented by azPx ) is balanced by
vertical diffusion (represented by kv Pzz ):
azPx kv Pzz .
(19)
The absolute fluid velocities are unimportant here—
only the difference in velocity across the layer
matters—therefore the appropriate vertical scale in
Eq. (19) is H min , the minimum layer thickness. Until
diffusion becomes important the horizontal profiles
of plankton concentration are translated in the
x-direction without modification and so the horizontal scale remains L0 (Fig. 5). Therefore, using
1
the scale estimates qz H 1
min and qx L0 , Eq. (19)
leads to the scaling
aH min
kv
2 ,
L0
H min
(20)
which may be rearranged to give the scaling of the
minimum layer thickness:
1=3
H min a1=3 k1=3
v L0 .
(21)
The minimum layer thickness in Eq. (21) depends
only on the shear a, the vertical diffusivity kv , and
the initial horizontal width of the patch L0 ; the
initial patch thickness H 0 is irrelevant to H min .
Stacey et al. (2007) contains an alternative derivation of Eq. (20) based on the slope of the layer. For
a patch with initial horizontal extent 1 km, a vertical
shear of 102 s1 , and a vertical diffusivity of
105 m2 s1 , the thinnest layers would be about
1 m thick. This is the same thickness as many of the
thin layers that have been observed in coastal waters
around the world (Dekshenieks et al., 2001;
Johnston et al., 2008).
We estimate tmin by combining Eq. (21) with the
approximate expression for HðtÞ in Eq. (17):
2=3
tmin a2=3 kv1=3 L0 .
(22)
Young et al. (1982), Rhines and Young (1983) and
Stacey et al. (2007) also derive Eq. (22).
The approximation for tmin in Eq. (22) shows that
the time required to achieve the minimum layer
thickness H min has no dependence on the initial
layer thickness H 0 . With the same shear and
diffusivities as above, a patch initially 1 km wide
would reach its minimum in about 1 day.
More precise estimates of H min and tmin in
Eqs. (A.28) and (A.26) show that the scale estimates
in Eqs. (21) and (22) are correct to within a factor of
31=6 1:2 and 31=3 1:4, respectively.
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The layer intensity IðtÞ changes only slightly
during the tilting and shear-thinning phases. Specifically, IðtÞ is given by Eq. (A.36), which in
dimensional variables is
1
IðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
1 þ 2ðt=tmin Þ3
(23)
When t5tmin the intensity is approximately 1 (see
Fig. 4). At the time of minimum thickness the
intensity is
1
Iðtmin Þ pffiffiffi .
3
(24)
Since 31=2 0:58, we see that despite thinning by a
factor of order 10 between t ¼ 0 and t ¼ tmin (as in
Fig. 4), the intensity decreases by less than a factor
of 2. Because only diffusion can change the
concentration of plankton in a water parcel in our
advection–diffusion model in Eq. (1), this modest
reduction in intensity is consistent with diffusion
not being important until t\tmin .
These scalings suggest that an initial patch of
plankton will form layers about 1 m thick over a
time scale of days, with very little decrease in
concentration during the shear thinning. The minimum layer thickness depends on the initial horizontal scale of the patch, the shear, and the vertical
diffusivity. The initial vertical scale and the horizontal diffusivity are irrelevant.
2.6. The decay phase
The decay phase occurs while
tmin ptptdiff ,
(25)
where tdiff is defined in Eq. (5). During this phase
the layer thickness slowly grows and the layer
intensity decays. During the decay phase the
approximation in Eq. (23) continues to describe
the reduction of IðtÞ. The exact solution for HðtÞ in
Eq. (A.18) may be simplified to give an approximate
expression for the layer thickness:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 t
1 tmin 2
HðtÞ H min
.
(26)
þ
3 tmin 3 t
The result above applies during both the shearthinning phase4 and the decay phase, and fails only
4
When t5tmin the approximation in Eq. (26) reduces to our
earlier expression for HðtÞ in Eq. (17).
at short times when totshear and at extremely long
times when t\tdiff btmin (Fig. 6).
2.7. The lifespan of a thin layer
The importance of a thin layer to the dynamics of
the planktonic ecosystem will depend on its duration. A layer that is transient relative to growth and
grazing rates will have less effect on the ecosystem
dynamics than one that persists for multiple
generations. Most phytoplankton have generation
times of about a day, and there is a dominant diel
periodicity to most trophic interactions; thin layers
that persist for days to weeks are likely to be a
significant local perturbation to the plankton
dynamics. To obtain a simple definition of the
lifetime of a thin layer we say that the layer is ‘‘thin’’
while
H min pHðtÞp2H min .
(27)
From (26) we find that the inequality above is
satisfied when
0:3tmin ptp6tmin .
(28)
Thus, according to the definition in Eq. (27), the
layer lifespan is approximately 5:7tmin . However,
toward the end of the period in Eq. (28), the layer
has suffered significant erosion: substituting t ¼
6tmin into Eq. (23) we find Ið6tmin Þ 0:05, or 5% of
the initial maximum. The point is that IðtÞ is much
greater during the shear-thinning phase than during
the decay phase. This asymmetry is a consequence
of the much greater horizontal extent of the layer
during the shear-dispersion phase. Thus Eq. (27) is
not a completely satisfactory definition of the
lifetime of a layer.
An alternative measure of layer lifetime is the
time from when the layer thins to twice the
minimum thickness until the layer intensity decays
to half the intensity at the time of minimum
thickness. With this alternative definition, the thinlayer lifetime is
0:3tmin ptp1:8tmin ,
(29)
so that the lifespan is 1:5tmin .
The main point here is that different definitions of
layer lifetime all lead to the same result: thin, intense
layers have lifespans of the same order as the time to
achieve minimum layer thickness, tmin . Under
typical oceanic conditions, we expect thin layers to
last several days to a week or so.
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H (m)
Tilting
Shear-thinning
Decay
287
Shear-dispersion
101
tmin
tshear
tdiff
100
L (m)
100
101
Tilting
103
t (hours)
tmin
tshear
107
102
104
105
tdiff
Shear-thinning
Decay
Shear-dispersion
105
103
100
101
102
103
104
105
t (hours)
Intensity
100
Tilting
Shear-thinning
Decay
Shear-dispersion
10-2
10-4
10-6
tshear
100
tmin
101
tdiff
102
103
104
105
t (hours)
Fig. 6. The layer thickness, extent, and intensity as a function of time, with the final time extended to show the shear-dispersion phase. The
final time on all three plots is 109 s (about 30 years) and the parameters are L0 ¼ 1000 m, H 0 ¼ 10 m, kh ¼ 1 m2 s1 , kv ¼ 105 m2 s1 , and
a ¼ 102 s1 . The vertical dashed lines are located at tshear , the end of the tilting phase defined in Eq. (4); tmin , the end of the shear-thinning
phase; and tdiff , the end of the shear-dispersion phase. (a) The layer thickness as a function of time. The ’s are the approximations for
layer thickness at the end of the tilting phase, at the approximate time of minimum layer thickness, and at the beginning of the sheardispersion phase. The dashed curve is the approximation in Eq. (26). (b) The horizontal layer extent as a function of time. The dashed
curve is the approximation in Eq. (12). (c) The layer intensity as a function of time—the change in slope from t3=2 to t2 may be seen at
t ¼ tdiff . The dashed curve is the approximation to the intensity in Eq. (23) using the approximation for tmin in Eq. (A.26).
2.8. The shear-dispersion phase
For the sake of completeness, we discuss the
shear-dispersion phase, during which shear dispersion destroys what little remains of the layer. We
define the shear-dispersion phase to be when
tdiff pt,
(30)
where the vertical diffusion time, tdiff , is defined in
Eq. (5). At this time
qffiffi
(31)
Hðtdiff Þ 35H 0 ,
and thus a main characteristic of shear dispersion is
that the layer thickness finally approaches and then
exceeds its initial value H 0 . A second important
characterization of the shear-dispersion phase is
that Eq. (12) finally fails and instead the horizontal
extent of the faint remnant grows like LðtÞt3=2 . The
shear-dispersion phase is not of much relevance to
the study of thin layers because
Iðtdiff Þ kv L0
51.
aH 30
(32)
Thus one might just as well refer to the sheardispersion phase as the oblivion phase. The
approximation for IðtÞ in Eq. (23) fails at
these very long times t4tdiff ; the intensity in the
shear-dispersion phase decreases faster than in
the decay phase, decreasing like t2 instead of
t3=2 [Fig. 6(c)].
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288
One point of interest is that during the sheardispersion phase H 2 ðtÞkv t=2, so that the ‘‘apparent
vertical diffusivity’’ is kv =4.
2.9. The horizontal extent of the layer
Unlike the layer thickness, the layer extent LðtÞ
increases monotonically during all four phases.
Eq. (12) applies during the first three phases and
begins to fail only when ttdiff and the sheardispersion phase begins. By evaluating the full
equation for layer extent, Eq. (A.31), and keeping
only the dominant terms, we find that the horizontal
pffiffiffi
extent of the layer increases by approximately 2
during the tilting phase,
pffiffiffi
Lðtshear Þ 2L0 .
(33)
At the end of the shear-thinning phase the layer
extent has increased to
1=3
3aL20
Lðtmin Þ H 0
.
(34)
kv
This is interesting because we see that the initial
vertical thickness affects the horizontal extent of the
layer. This is because the greater the distance
between the top and bottom of the layer, the greater
the difference in velocities experienced by the edges
of layer and the farther they will separate.
During the final shear-dispersion phase, when
tdiff 5t, Eq. (12) is replaced by
qffiffi
3=2
LðtÞ 23ak1=2
.
(35)
v t
The t3=2 growth of L is characteristic of shear
dispersion in an unbounded region (e.g., Rhines and
Young, 1983).
3. Plankton growth
Plankton are of course not inert tracers, and so
we consider here a model of an initial patch of
nutrients (perhaps due to upwelling or a breaking
internal wave injecting nutrients into the euphotic
zone) and show how the similarity of plankton
growth time scales and the time to minimum layer
20
8
6
0
4
tshear
–10
tmin
2
–20
0
0
20
40
60
t (hours)
80
100
120
20
2.5
10
z (m)
2
0
tshear
–10
1.5
tmin
P (arbitrary units)
z (m)
10
N (arbitrary units)
10
1
–20
0
20
40
60
t (hours)
80
100
120
Fig. 7. Vertical profiles of N and P from the system in Eqs. (36) and (37) at x ¼ 0 m as a function of time. The parameters are
L0 ¼ 1000 m, H 0 ¼ 10 m, kh ¼ 1 m2 s1 , kv ¼ 105 m2 s1 , a ¼ 102 s1 , r ¼ 105 s1 , and N ¼ 2pL0 H 0 N ¼ 10 (arbitrary units). The
dashed black curves are HðtÞ. (a) Nð0; z; tÞ. The reduction in vertical scale due to shear thinning is readily apparent for totmin and then
the nutrients are consumed. (b) Pð0; z; tÞ. The plankton is thinning for t5tmin , but it is hard to see because the plankton intensity is low and
then around t ¼ tmin =2 an intense thin layer appears.
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D.A. Birch et al. / Deep-Sea Research I 55 (2008) 277–295
thickness may cause a thin layer of plankton to
appear as if out of nowhere (Fig. 7). The system we
consider is the NP model
Pt þ azPx ¼
rNP
þ kh Pxx þ kv Pzz ,
N
N t þ azN x ¼ (36)
rNP
þ kh N xx þ kv N zz ,
N
(37)
where the phytoplankton P has a net growth rate
rN=N when the available nutrient concentration is
N. Typical values of rN=N in a patch of nutrients
are around 105 s1 ; somewhat coincidently, the
time for the plankton to use up the patch of
nutrients and grow into an intense feature is similar
to tmin , the time to reach the minimum layer
thickness. The dimensions of r are s1 and the
dimensions of N and N are arbitrary so long as one
unit of N can be converted by plankton growth into
one unit of P.
For an initial condition we let the concentration
of phytoplankton equal a small background level P0
and take a Gaussian patch with total nutrients N as
the initial nutrient profile
Pðx; z; 0Þ ¼ P0 ,
(38)
!
N
x2
z2
Nðx; z; 0Þ ¼
exp 2 .
2pL0 H 0
2L0 2H 20
(39)
We connect to the preceding sections by observing
that the quantity N þ P P0 will behave exactly
like the plankton patch in Eq. (1) with the initial
condition in Eq. (2) and experience all four phases
of layer evolution. When we refer to the layer
thickness, we really mean the thickness of the
anomaly, P P0 , and the linearity of Eq. (1)
allowed us to ignore the background plankton
concentration P0 . In a model with growth, such as
Eqs. (36) and (37), the background concentration is
important since phytoplankton must be present to
take advantage of a patch of nutrients.
In Appendix B we show that the vertical thickness
of the plankton anomaly in the system Eqs. (36) and
(37) is the same as the vertical thickness of the layer
in the original model in Eq. (1). At times less than
tmin the plankton distribution in the NP model is
thinning as the phytoplankton grow, but the
intensity is still low. When measured in the field,
this layer might not be considered an interesting
feature until t approaches tmin and the phytoplankton layer becomes more intense. At times much
greater than tmin the plankton will have used up all
the nutrients and the plankton anomaly, P P0 , in
the NP model evolves exactly like P in Eq. (1). The
lack of an obvious phytoplankton signal (above the
background) during the tilting phase and early part
of the shear-thinning phase causes the phytoplankton layer to appear de novo as a thin layer, even
though the nutrient patch that caused the layer had
a larger initial vertical extent.
4. Summary and discussion
To summarize the range of time scales and
thicknesses of thin layers we solve Eq. (A.18)
numerically to find the minimum layer thickness
H min and the time to minimum layer thickness tmin
as a function of the initial horizontal extent L0
and the shear a, holding the other parameters fixed
(Fig. 8). We take kh ¼ 1 m2 s1 , kv ¼ 105 m2 s1
because these values of diffusivity are the same
100
10-2
100
1
10 h
10-1
102 h
α (s-1)
α (s-1)
10-1
3
10 h
10-3
10-4
102
289
5-1 m
0
5 m
10-2
51 m
10-3
10-4
103
104
L0 (m)
105
102
103
104
105
L0 (m)
Fig. 8. (a) The time to minimum layer thickness tmin as a function of the initial horizontal width L0 and the shear a. (b) The minimum layer
thickness H min as a function of the initial horizontal width L0 and the shear a. These results are obtained by solving Eq. (A.25) numerically
for tmin and substituting the results into Eq. (A.18). The parameters in both panels are kh ¼ 1 m2 s1 , kv ¼ 105 m2 s1 , and H 0 ¼ 10 m.
The shaded regions are where H 0 =L0 is too small and no shear thinning occurs.
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order as the horizontal and diapycnal diffusivities found by Sundermeyer and Ledwell (2001),
MacKinnon and Gregg (2003), Ledwell et al. (2004),
and Oakey and Greenan (2004). From our earlier
analyses we have shown that the values of kh and
H 0 are not relevant in the determination of the layer
thickness, so long as H 0 is not so small that the layer
1=3 1=3
does not thin. If H 0 o31=6 a1=3 kv L0 , then the
layer starts as thin as it will ever be and thickens
monotonically. Overall, we find that for oceanic
parameters, constant shear can produce layers
about 1 m thick in a time of order 1 day. Weaker
shears produce layers that tilt and thin over periods
of weeks before diffusing away. Stronger shears can
produce thinner layers with shorter lifetimes in less
than 1 day.
The evolution of the steady-shear model in
Eq. (1) with Gaussian initial condition in Eq. (2)
consists of four phases: the tilting phase, the shearthinning phase, the decay phase, and the sheardispersion phase. During the tilting phase the initial
patch rotates and stretches in the sheared flow, with
little change in its vertical thickness. The tilting
phase lasts from t ¼ 0 to t ¼ tshear L0 =aH 0 , and
the player
thickness at tshear is approximately
ffiffiffi
H 0 = 2 0:7H 0 .
After the tilting phase, the layer enters the shearthinning phase, which lasts until t ¼ tmin 1=3 2=3
31=3 a2=3 kv L0 . During the shear-thinning phase
the layer thickness decreases like t1 , with
HðtÞL0 =at, until reaching the minimum thickness,
1=3 1=3
H min 31=6 a1=3 kv L0 at tmin . It is at this time
that diffusion balances shear thinning and the layer
begins to thicken and decay in intensity. Interestingly, the most-studied phase, the shear-dispersion
phase, turns out to be irrelevant to thin layers: by
the time the shear-dispersion phase begins, the layer
intensity has decayed to an insignificant level and
the layer thickness is comparable to the initial patch
thickness. We also emphasize the importance of
initial conditions: just finding the Green’s function
for Eq. (1) does not reveal the full behavior of the
model.
A curious result of our analyses is that the
thickness of the layer does not depend on the initial
vertical extent of the patch, but rather its horizontal
scale. For patches initially 1 km or so wide, vertical
shear will create layers 1 m thick that persist for
days to weeks. These time scales are long enough to
be biologically relevant: herbivorous zooplankton
could find and exploit such layers; the enhanced
concentrations could lead to increased sexual
exchange or infection within the layers; and particle
collisions within high-biomass layers of certain
phytoplankton types could lead to aggregation,
sinking, and locally episodic carbon fluxes out of the
euphotic zone.
The simple NP model gives some interesting
insights into the time- and space-dependence of
phytoplankton layers that might be observed in the
field. A nutrient injection to the euphotic zone is
sheared as the phytoplankton are growing. This
leads not to a thick patch of phytoplankton, but
rather the growth of the phytoplankton in an
initially thin layer. This is a consequence of the
similarities in the time scales for layer tilting and
thinning, and phytoplankton growth. By the time
the phytoplankton have reached a concentration
significantly above the background, the initial patch
of nutrients has already undergone significant tilting
and thinning. It will be interesting, given the recent
availability of technologies such as the in situ
ultraviolet spectrophotometer (ISUS) optical nitrate
sensor (Satlantic, Halifax NS, Canada), to explore
these predicted dynamics in the field. Regions such
as wind-driven upwelling systems, or flow around
topography are likely sites for nutrient injections to
the euphotic zone with subsequent phytoplankton
growth and shearing.
Our model assumes an initial patch of phytoplankton or nutrients with a finite vertical and
horizontal extent. Such a patch could form through,
for example, nutrient injection in a coastal winddriven upwelling system (Johnston et al., 2008),
nutrient injection through breaking internal waves,
upwelling and patch formation in fronts, meanders,
and eddies, or local wind-driven mixing under a
storm track. While we have concentrated on the
formation of layers of locally enhanced biomass, it
is important to recognize that the model applies to
any patch of a property. Thus a local patch of a
particular community or species of plankton with a
biomass indistinguishable from the surrounding
community will form a thin layer in a verticallysheared current. Such layers of properties—even in
the absence of any bulk concentration gradients
such as chlorophyll concentration—should be ubiquitous features in the ocean.
The simple model in Eq. (1) contains no
biological processes; it quantifies for any tracer
(e.g. plankton species, plankton community, chemical) Eckart’s observation that sheared currents will
transform horizontal variability into vertical variability. Given the existence of sheared currents and
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D.A. Birch et al. / Deep-Sea Research I 55 (2008) 277–295
horizontal variability (‘‘patchiness’’), vertical layers
in the ocean are unavoidable.
The layers described here have several diagnostics
that should make them detectable in the field. Most
importantly, layers formed by vertical shearing of a
patch should incline across isopycnals, as most of
the vertical shear in the ocean is perpendicular to
isopycnal surfaces. Over the horizontal extent LðtÞ
of the layer, the local maximum in intensity will be
found on different isopycnals, with one end of the
patch in lower-density water, and the opposite end
in higher-density water. Vertical shear requires
vertical stratification: if the stratification is too
weak, the shear will cause turbulent mixing which
will erode a layer. Thus we expect thin layers to
form in stratified waters, with the thinnest (and
shortest-lived) layers forming in the regions of
strongest stratification and shear (Fig. 8). Finally,
we predict that as technologies are developed,
extensive vertical structure of most chemical and
biological properties in the ocean will be revealed.
Shear is ubiquitous in the ocean, horizontal patchiness is constantly being created, and so thin layers
of properties should be expected, rather than
surprising.
Acknowledgments
The authors would like to thank Jennifer
MacKinnon, Rob Pinkel, and Shaun Johnston for
helpful discussions. This work was supported by
National Science Foundation Grants OCE-0220362
and OCE-0726320 to W.R.Y., and NSF Grant
OCE-0220379 and ONR Grant N00014-06-1-0304
to P.J.S.F.
Appendix A. Gaussian with the method of moments
291
of plankton at all times:
Z 1Z 1
Pðx; z; tÞ dx dz ¼ P.
1
(A.2)
1
We define the second moments5 of the plankton
distribution using the global average defined in
Eq. (A.1):
m11 0x2 T;
m22 0z2 T;
m12 0xzT.
(A.3)
We obtain the three second moments of the
plankton distribution by applying the same method
of moments as Aris (1956) with the initial conditions
m11 ð0Þ ¼ L20 ;
m12 ð0Þ ¼ 0;
m22 ð0Þ ¼ H 20 .
2
(A.4)
2
Multiplying Eq. (1) by x , xz, and z and averaging
according to Eq. (A.1) yields a set of coupled
equations for the second moments which may be
solved to obtain
m11 ðtÞ ¼ L20 þ 2kh t þ a2 H 20 t2 þ 23a2 kv t3 ,
(A.5)
m12 ðtÞ ¼ aH 20 t þ akv t2 ,
(A.6)
m22 ðtÞ ¼ H 20 þ 2kv t.
(A.7)
Armed with all of the first and second moments of
the plankton distribution we can write down the
solution to Eq. (1) with the initial condition in
Eq. (2) (van Kampen, 1981)
P
1
Pðx; z; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp xT M 1 ðtÞx , (A.8)
2
2p MðtÞ
where x is a column vector and M is the moment
matrix with determinant M:
!
m11 ðtÞ m12 ðtÞ
x
x
; MðtÞ (A.9)
m12 ðtÞ m22 ðtÞ
z
and
In this Appendix we present the complete
solution to Eq. (1) with initial condition Eq. (2).
We also formally define the layer properties,
introduce a convenient non-dimensionalization,
and derive approximations to the layer properties
during the four phases of layer evolution.
First, define the global average of a function f
weighted by the distribution Pðx; z; tÞ:
R1 R1
1 fP dx dz
0f T R1
.
(A.1)
1 R1
1 1 P dx dz
A useful consistency check is to note that 01T ¼ 1.
Also, note that Eq. (1) conserves the total amount
MðtÞ det MðtÞ ¼ m11 m22 m212 .
(A.10)
The moment matrix MðtÞ is real and symmetric and
its inverse M 1 ðtÞ appearing in Eq. (A.8) is
!
m22 ðtÞ m12 ðtÞ
1
1
M ðtÞ ¼
.
(A.11)
MðtÞ m12 ðtÞ m11 ðtÞ
Thus we have now found the complete solution to
Pðx; z; tÞ. Next we use this solution to examine the
layer thickness.
The first moments, 0xT and 0zT, are both initially zero and
remain zero for all time.
5
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292
A.1. Layer thickness and extent
To examine properties of the distribution Pðx; z; tÞ
in the vertical, such as layer thickness, we must
define an average over z only
R1
fP dz
hf i R1
.
(A.12)
1
1 P dz
To evaluate quantities such as hzi we use the
expression for Pðx; z; tÞ in Eq. (A.8) and complete
the square in the exponential to obtain
n
Z 1 rffiffiffiffiffiffiffiffi
2
1
2M
m12
n
hz i ¼ pffiffiffi
x ez dz.
(A.13)
zþ
m11
m11
p 1
Specifically, the center of mass of any vertical
profile is
hzi ¼
m12
x.
m11
(A.14)
Similarly,
hz2 i ¼
m212
m211
x2 þ
M
.
m11
(A.15)
(A.16)
Use Eqs. (A.14) and (A.15) to find the square of the
thickness:
H 2 ðx; tÞ ¼
M
m2
¼ m22 12 ,
m11
m11
(A.19)
Furthermore, we define the non-dimensional time t
and non-dimensional thickness h:
t Eat;
h
H
.
EL0
(A.20)
Now express the equation for layer thickness,
Eq. (A.18), in non-dimensional variables
h2 ðtÞ ¼
E2 t4 þ 2Z2 t3 þ 12E4 wt2 þ 6E2 ð1 þ Z2 wÞt þ 3Z2
.
2E2 t3 þ 3Z2 t2 þ 6E4 wt þ 3E2
(A.21)
We obtain an approximate equation for the layer
thickness as a function of time by dropping all of
the terms containing E from the right-hand side of
Eq. (A.21)
2t3 þ 3
.
(A.22)
3t2
Implicit in this simplification is the assumption
that
h2 ðtÞ Define the square of the thickness H of any vertical
profile to be
H 2 ðx; tÞ hz2 i hzi2 .
three dimensionless quantities:
!1=3
kv
H0
kh
; Z
; w .
E
L0
kv
aL20
(A.17)
or we can save some algebra by observing that
Eq. (A.8) shows that for the Gaussian model, the
square of the formal layer thickness is the same as
1=M 1
22 . Note that Eq. (A.17) shows that Hðx; tÞ is
independent of x for our Gaussian distribution, and
so we will neglect the x from now on. To obtain an
explicit expression for HðtÞ in terms of t and the
parameters of the model, substitute Eqs. (A.7),
(A.6), and (A.5) into Eq. (A.17):
a2 k2 t4 þ 2a2 H 20 kv t3 þ a2 H 40 t2
H 2 ðtÞ ¼ H 20 þ 2kv t 2 2 v 3
.
2
2 2 2
3 a kv t þ a H 0 t þ 2kh t þ L0
(A.18)
Using the solution for HðtÞ in Eq. (A.18), we can
now find the minimum layer thickness and the time
to minimum layer thickness. However, before
proceeding any farther, it is advantageous to switch
to dimensionless variables. Therefore, we define
w5E2 ,
(A.23)
kh =L20
in the
which corresponds to ignoring the term
scaling argument in Eq. (20).
Let tmin be the time of minimum layer thickness.
Then
d½h2 ¼ 0,
(A.24)
dt t¼tmin
or equivalently,
t4min 3tmin 0.
(A.25)
We ignore the root tmin 0 and take the only
remaining real root, tmin 31=3 . Using the
definition of t in Eq. (A.20) we find tmin , the time
of minimum layer thickness, in dimensional variables
2 1=3
1=3 1 aL0
tmin 3 a
.
(A.26)
kv
To find an approximation for H min , the minimum layer thickness, substitute t ¼ 31=3 into
Eq. (A.22),
hmin 31=6 .
(A.27)
Using Eq. (A.27) and the definition of E in
Eq. (A.19) yields an approximation to the minimum
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D.A. Birch et al. / Deep-Sea Research I 55 (2008) 277–295
layer thickness in dimensional variables
!1=3
kv
1=6
H min 3 L0
.
aL20
Using the definitions of E, Z, and w and writing t as
a1 E1 t one may rewrite Eq. (A.34) as
(A.28)
We would also like to know the time at which the
layer has a particular thickness. To find t as a
function of H we rearrange the equation for layer
thickness, Eq. (A.22), as a cubic polynomial in
time
2t3 3h2 t2 þ 3 ¼ 0.
(A.29)
1=6
If h is large (and 2 3 is large enough), then the
two positive roots of Eq. (A.29) are
1
3h2
and
.
(A.30)
h
2
Both roots in Eq. (A.30) are times when the layer
thickness is 31=6 hH min ; the first is during the shearthinning phase and the second is during the sheardispersion phase. Note that when tbtmin then the t4
term in Eq. (A.21) is important and Eq. (A.29) and
its roots Eq. (A.30) are incorrect. Appendix A.3
contains an approximation for HðtÞ for very large
times.
We define the horizontal extent of the layer LðtÞ
to be
pffiffiffiffiffiffiffiffi
LðtÞ m11 .
(A.31)
t
Note that the horizontal analog of HðtÞ in
Eq. (A.17) is m11 m212 =m22 , which is the horizontal
thickness (as opposed to the horizontal extent) of
the layer. However, we choose not to examine m11 m212 =m22 because we focus on quantities observed by
vertical (not horizontal) profiles.
A.2. Layer intensity
To obtain an equation for the layer intensity we
substitute the solution for P, Eq. (A.8), into the
definition of I, Eq. (3):
Pð0; 0; tÞ
Pð0; 0; 0Þ
sffiffiffiffiffiffiffiffiffiffiffi
Mð0Þ
¼
MðtÞ
"
!
2kv 2kh
4kv kh
¼ 1þ
þ 2 t þ 2 2 t2
H 20
L0
H 0 L0
#1=2
2a2 kv 3
a2 k2v 4
þ
t þ
t
.
2
3L0
3H 20 L20
I
293
ðA:32Þ
ðA:33Þ
IðtÞ ¼ ½1 þ 2E2 ðZ2 þ wÞt þ 4E4 Z2 wt2
þ 23t3 þ 13E2 Z2 t4 1=2 .
Neglecting all terms containing E in Eq. (A.35)
yields a simple equation for the approximate layer
intensity as a function of time (as long as t is large
enough so that the t3 term is larger than the t1 term,
but not so large that the t4 is relevant)
1
IðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
1 þ 23 t3
(A.36)
Substituting the value t ¼ 31=3 (t ¼ tmin ) into
Eq. (A.36) one obtains the layer intensity at the
approximate time of minimum layer thickness:
Iðtmin Þ 31=2 .
(A.37)
At t ¼ tshear , the end of the tilting phase, tshear ¼
EZ1 and Eq. (A.36) says that the layer intensity is
indistinguishable from 1.
We can also find time as a function of intensity by
inverting Eq. (A.36):
2
1=3
3I 3
t
.
(A.38)
2
Eq. (A.38) has one real root, which is consistent
with the layer intensity monotonically decreasing
with time.
A.3. Long-time behavior and the shear-dispersion
phase
The approximations for layer thickness in
Eqs. (A.30) and intensity (A.36) are only good
when t in Eq. (A.20) is not too large; eventually
the t4 terms in the full equations [Eqs. (A.21)
and (A.35)] dominate and new approximations
for the layer thickness and intensity become
necessary. The t4 terms become important when
they are of the same order as the t3 terms in
Eqs. (A.21) and (A.35). This happens when ttdiff
or equivalently
Z2
.
(A.39)
E2
Substituting Eq. (A.39) into the full equations for
thickness, Eq. (A.21), we find
rffiffiffi
3Z
,
(A.40)
hðtdiff Þ 5E
t ¼ tdiff ðA:34Þ
ðA:35Þ
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294
which means that before the t4 terms dominate and
Eqs. (A.44) and (A.42) apply, the layer is almost as
thick as when it started. Similarly, the full equation
for layer intensity, Eq. (A.35), shows that
Iðtdiff Þ E3
51
Z3
(A.41)
Pðx; z; tÞ ¼
and the layer intensity is tiny before Eqs. (A.44) and
(A.42) apply.
For the sake of completeness we note the long
time behavior of H, L, and I:
t
(A.42)
h2 ðtÞ ; t ! 1,
2
qffiffi
LðtÞ 23L0 E3=2 t3=2 ; t ! 1,
(A.43)
pffiffiffi Z
IðtÞ 3 2 ;
Et
where x, M, and M are the same as in Eqs. (A.9)
and (A.10). We obtain an approximation for P
during the tilting and shear-thinning phases by
neglecting diffusion in Eqs. (36) and (B.2), in which
case we can write
t ! 1.
(A.44)
Finally, we note two interesting properties (both
may be found in the d-function solution of Novikov,
1958) of the long-time behavior shear-dispersion
phase. First, the layer thickness is proportional to
t1=2 , as one would see with diffusion alone, but
because of the shear the effective diffusion coefficient is one-quarter kv . Second, in a purely diffusive
system the layer intensity would behave as t1 , but
in this system the intensity at long times is
proportional to t2 .
Appendix B. Plankton growth
In this appendix we derive an approximation for
the short-time behavior of the vertical thickness of
the plankton anomaly in the NP system [Eqs. (36)
and (37)]. First we define
Q N þ P.
(B.1)
Q satisfies the exact same equation as P in Eq. (1),
Qt þ azQx ¼ kh Qxx þ kv Qzz ,
(B.2)
with a slightly different initial condition,
!
N
x2
z2
Qðx; z; 0Þ ¼ P0 þ
exp 2 .
2pL0 H 0
2L0 2H 20
(B.3)
The solution for Q is
Qðx; z; tÞ ¼ P0 þ
N
1
pffiffiffiffiffiffiffiffiffiffiffi exp xT M 1 ðtÞx ,
2
2p MðtÞ
(B.4)
P0 Qðx; z; tÞ
,
P0 þ ðQðx; z; tÞ P0 Þ exp½rQðx; z; tÞt=N (B.5)
Qðx; z; tÞ ¼ Qðx azt; z; 0Þ.
(B.6)
We are really interested in the growth of plankton
anomaly,
P0 ðx; z; tÞ
Pðx; z; tÞ P0
¼
P0 ½Qðx; z; tÞ P0 ð1 exp½rQðx; z; tÞt=N Þ
,
P0 þ ½Qðx; z; tÞ P0 exp½rQðx; z; tÞt=N ðB:7Þ
which for small t (which is also when we can neglect
diffusion) may be approximated by
x2
½z SðtÞx2
0
P ðx; z; tÞ=P0 rt exp 2 ,
2L ðtÞ
2H 2 ðtÞ
(B.8)
with HðtÞ, LðtÞ, and SðtÞ defined in Eqs. (11), (12),
and (13). Therefore we have shown that the NP
model forms layers with the same thickness as the
inert tracer model, Eq. (1).
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